On the Value of Job Migration in Online Makespan Minimization

Abstract

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of n jobs has to be scheduled non-preemptively on m machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range [1.88, 1.9201]. Currently no randomized online algorithm with a smaller competitiveness is known, for general m. In this paper we explore the power of job migration, i.e. an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is \(\alpha _m\)-competitive, for any \(m\ge 2\), where \(\alpha _m\) is the solution of a certain equation. For \(m=2\), \(\alpha _2 = 4/3\) and \(\lim _{m\rightarrow \infty } \alpha _m = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2)) \approx 1.4659\). Here \(W_{-1}\) is the lower branch of the Lambert W function. For \(m\ge 11\), the algorithm uses at most 7m migration operations. For smaller m, 8m to 10m operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses o(n) job migrations can achieve a competitive ratio smaller than \(\alpha _m\). We finally trade performance for migrations. We give a family of algorithms that is c-competitive, for any \(5/3\le c \le 2\). For \(c= 5/3\), the strategy uses at most 4m job migrations. For \(c=1.75\), at most 2.5m migrations are used.

Keywords

A preliminary version of this paper has appeared in Proc. 20th Annual European Symposium on Algorithms (ESA), 2012. Susanne Albers’ work supported by the German Research Foundation, project Al 464/7-1.

Substituting \(x=1/(\alpha -1)\), which is equivalent to \(\alpha =1/x+1\), we find that the above is equivalent to \(xe+e = e^x\). Applying the Lambert W function we find that \(x=-W_{-1}(-1/e^2)-1\) is a solution of the former equality. Substituting we conclude that in fact \(\alpha = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2))\) satisfies the equality. Using the same techniques we can show that \(\lim _{m\rightarrow \infty } \alpha _m\) is lower bounded by \(W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2))\). In the calculations, (1) yields that \(H_{m-1} - H_{\lceil cm \rceil } < \ln (1/c) + 1/(2m)\). \(\square \)